What I want to go

over in this video is a special case

of L’Hopital’s Rule. And it’s a more

constrained version of the general case

we’ve been looking at. But it’s still very powerful

and very applicable. And the reason why we’re going

to go over this special case is because its proof is

fairly straightforward and will give you an intuition

for why L’Hopital’s Rule works at all. So the special case

of L’Hopital’s Rule is a situation where

f of a is equal to 0. f prime of a exists. g of a is equal to 0. g prime of a exists. If these constraints

are met, then the limit, as x approaches a of

f of x over g of x, is going to be equal to f

prime of a over g prime of a. So it’s very similar

to the general case. It’s little bit

more constrained. We’re assuming that

f prime of a exists. We’re not just

taking the limit now. We’re assuming f prime of a and

g prime of a actually exist. But notice if we substitute

a right over here we get 0/0. But that if the

derivatives exist we could just evaluate

the derivatives at a, and then we get the limit. So this is very close

to the general case of L’Hopital’s Rule. Now let’s actually prove it. And to prove it, we’re going

to start with the right hand and then show that if we use

the definition of derivatives, we get the left hand

right over here. So let me do that. So I’ll do it right over here. So f prime of a

is equal to what, by the definition

of derivatives? Well, we could view

that as the limit as x approaches a of f of x

minus f of a over x minus a. So this is literally just

a slope between two points. So like, if you have your

function f of x like this, this is the point a, f

of a right over here. This right over here

is the point x, f of x. This expression

right over here is the slope between

these two points. The change in our y value

is f of x minus f of a. The change in our f

value is x minus a. So this expression is just

the slope of this line. And we’re just taking

the– let me actually do that in a different

color– the line that connects these two points,

that’s the slope of it. I’ll do that in white. The slope of the line that

connects those two points. And we’re taking

the limit as x gets closer and closer

and closer to a. So this is just

another way of writing the definition of

the derivative. So that’s fine. Let’s do the same

thing for g prime of a. So f prime of a

over g prime of a, is going to be this

business which is in orange, f prime of a over g prime of a. Which we can write as

the limit as x approaches a of g of x minus g

of a over x minus a. Well, in the numerator,

we’re taking the limit as x approaches a, and

in the denominator, we’re taking the limit

as x approaches a. So we can just rewrite this. This we can rewrite as

the limit as x approaches a of all this

business in orange. f of x minus f of

a, over x minus a, over all the business in green. g of x minus g of a, all

of that over x minus a. Now, to simplify this, we

can multiply the numerator and the denominator by x minus

a to get rid of these x minus a’s. So let’s do that. Let’s multiply by x

minus a over x minus a. So the numerator, x minus a,

and we’re dividing by x minus a. Those cancel out. And then these two cancel out. And we’re left with this thing

over here is equal to the limit as x approaches a

of, in the numerator we have f of x minus f of a. And in the denominator, we

have g of x minus g of a. And I think you see

where this is going. What is f(a) equal to? Well, we assumed f

of a is equal to 0. That’s why we’re

using L’Hopital’s Rule from the get go. f of a is equal

to 0, g of a is equal to 0. f of a is equal to 0.

g of a is equal to 0. And this simplifies to the

limit as x approaches a of f prime of x, sorry of f of

x, we’ve got to be careful. Of f of x over g of x. So we just showed that if f of

a equals 0, g of a equals 0, and these two derivatives exist,

then the derivatives evaluated at a over each other are

going to be equal to the limit as x approaches a of

f of x over g of x. Or the limit as x

approaches a of f of x over g of x is going to

be equal to f prime of a over g prime of a. So fairly straightforward

proof for the special case– the special case, not

the more general case– of L’Hopital’s Rule.

Hi Sal. You did a proof of the Fundamenta theorem of Calculus, but relied there on mean value theorem, which there's no video with proof of. Could you please make a video on that? thank you.

huh

good …

look, i never said that i didn't have the intuition. The intuition is there, and it's pretty strong. But having the intuition is one thing, and having a rigorous proof is another. When you learn mathematics, you should feel like a judge. And a judge seldom makes conclusions based on intuition, rather, he would look for a convincing argument. What intuition does is points out a way to go to find the argument.

Slight nitpick: I think you need to say that g'(a) must be non-zero.

I actually like L'hospital's Rule, It does simplify finding the limit

@1:04 gr8 choice of colors , sal. 🙂

the colors are exactly in according to the Indian flag !! 😀

What does he mean by saying with respect to x and y or xy

Wow.

I was surfing for boobs and ended up here in the weird part of youtube again.

I would go to bed but this is fucking spectacular! So…

If F(a) is the left boob and G(a) is the right boob then F'(a) / G'(a) = naked woman lying on her right side.

Mind blown. Time for bed, y'all. I'm going to be the numerator and my fucking mattress is going to be my denominator. It's your denominator too, niggaz! G'night.

I looooooove you

Hello. Great video. Could you please tell me where I can find this special case of l'Hopital rule written formally. Maybe on some official calculus paper or something like that?

Sir can you make a video on explaining l hopitals rule on sums like"log(log(sin(tan(x^2-2x)^3", this is just an example, but sums like this !?!?

Confused ! So I assume that one has to prove the limit exists before one can take it, which begs the question…….how can this be done without actually taking the limit? This appears to be a contradiction in terms, or at least ambiguous. Any clarification would be appreciated and thanks for great insightful videos.

what is the proof behind infinity/infinity?

Well .. some indefinities here and there but it's ok , thats not a video for mathematicians . Great job for sharing knowledge with the world !!!

Check out [email protected]

Sal. Khan, I wish u were a professor at my university :/

Hopital should be afraid from you!!! 🙂

Why do we use the mean value theorem rather than the definition of a derivative?

Where do calc students go when they get sick?

L'Hopital

What if f(a)=infinity?How can you derive it?